3.33 PDE change of variables?

Given pde \(y \left (\frac {\partial }{\partial y}u \left (x , y\right )\right )+x \left (\frac {\partial }{\partial x}u \left (x , y\right )\right ) = 0\) This below changes the pde to polar coordinates

pde:=y*diff(u(x,y),y)+x*diff(u(x,y),x)=0; 
the_tr:={x=r*cos(t),y=r*sin(t)}; 
PDEtools:-dchange(the_tr,pde,[r,t]);
 

\[ \sin \left (t \right ) \left (\left (\frac {\partial }{\partial r}u \left (r , t\right )\right ) \sin \left (t \right ) r +\cos \left (t \right ) \left (\frac {\partial }{\partial t}u \left (r , t\right )\right )\right ) -\cos \left (t \right ) \left (-\cos \left (t \right ) \left (\frac {\partial }{\partial r}u \left (r , t\right )\right ) r +\left (\frac {\partial }{\partial t}u \left (r , t\right )\right ) \sin \left (t \right )\right ) = 0 \]